A New Method for Determining the Pattern Speed and the Star Formation Timescale of Spiral Galaxies

Transcription

1 A New Method for Determining the Pattern Speed and the Star Formation Timescale of Spiral Galaxies Fumi EGUSA 0 Institute of Astronomy, The University of Tokyo, Osawa, Mitaka, Tokyo , Japan Thesis for the Master s Degree Submitted: January 31, 2005 Revised: March 8, 2005 ABSTRACT We developed a new method for the determination of the pattern speed (Ω P ) and star formation timescale (t SF ) in spiral galaxies. Though the pattern speed is a fundamental parameter, which governs kinematics and morphology of a galactic disk, its value cannot be derived directly from the observations. Our method uses azimuthal offsets between arms of massive stars and molecular gases. Assuming a circular rotation, this offsets (θ) should exhibit a linear dependence on the angular rotation velocity of materials (Ω), expressed by θ Ω Ω P t SF. With this relation, we can determine Ω P and t SF simultaneously by fitting a line to θ Ω plot, if we assume they are constant. We applied this method to three late type spiral galaxies, NGC 4254, NGC 4321, and NGC We used CO images for molecular distributions, and Hα and B-band images for massive star distributions. We found a clear linear relationship between the offsets and angular rotation velocities for NGC 4254 and NGC This is the first work to show this relationship, and its linearity verifies that the assumptions we used in our method are reasonable. For NGC 4254, 17 we obtained t SF Myr and Ω P 26 8 km s 1 kpc 1. For NGC 5194, we obtained 4 t SF Myr and Ω P 19 3 km s 1 kpc 1. The resultant values of t SF 4 Myr indicate that the gravitational collapse might be a main mechanism of star formation. The result of the rest galaxy, NGC 4321, in which we could not identify offsets, suggests that the extinction in B- band light is significant in this galaxy and that, offset values from CO and B-band data could be less reliable. We emphasize that this method is based on simple and reasonable assumptions and enables us to determine Ω P and t SF simultaneously and directly from the observations. 0 fegusa@ioa.s.u-tokyo.ac.jp

2 2 Contents 1 Introduction Pattern Speed Density Wave Theory Previous Methods Molecular Clouds and Star Formation Galactic Shock-Wave Theory Observational Tracers Method Basic Idea and Formulation Requirements for the Application Data NGC NGC 4321 and NGC Process of the Application Phase Diagram Azimuthal Intensity Profile θ Ω Plot Errors Fitting Results NGC NGC

3 3 5.3 NGC Discussion Summary 42

4 4 1. Introduction 1.1. Pattern Speed Density Wave Theory When we observe extragalaxies, we notice that there are so many types of galaxies. An American astronomer, Edwin Hubble classified them by their morphologies. His classification consists of three main categories, elliptical (E), lenticular (S0), and spiral galaxies (S), which form a single sequence: E S0 Sa Sb Sc Sd. Galaxies near the start of this sequence are called early type galaxies and have little gas and dust, while those near the end are called late type and rich in gas, dust, and young stars. Some spiral and lenticular galaxies have a central bar, and they form another sequence SB0 SBa SBb SBc SBd, which is parallel to that of the unbarred galaxies. Early type spirals are often brighter and exhibit bigger bulges than late type spirals. Their arms tend to be more tightly wound. The arm structure is often interpreted to be a logarithmic function as lnr tanξ φ (1) while R is a galactocentric radius, ξ is a pitch angle, and φ is an azimuthal angle. The pitch angle is defined as an angle between the tangential lines of a circle with radius R and the spiral arm (figure 1). If the disk rotates in the same direction with φ, the arms are leading. If the disk rotates in the opposite direction with φ, the arms are trailing. Observations of the nearest galaxies suggest that most spirals are trailing. Kennicutt (1981) measured pitch angles for Sa-Sc galaxies, and showed that an average pitch angle of late type galaxies was larger than that of early types. In other words, the arms of late types are more open, while those of early types are more closed. It is very consistent to the Hubble sequence described above. As the Hubble s classification has been modified, spiral galaxies are now defined to be galaxies with a morphological T type from 1 to 8, corresponding to types of S(B)a, S(B)ab,..., S(B)d, and S(B)dm. If we look up a catalog of galaxies, we can find that about half or more of nearby galaxies fall in these categories. Tully (1988) compiled 2367 galaxies in Nearby Galaxies Catalogue, and 1171 (49%) galaxies are spirals. In Third Reference Catalogue of Bright Galaxies (de Vaucouleurs et al. 1991, hereafter RC3), there are galaxies, and among them, (76%) galaxies are classified as spirals. Elmegreen & Elmegreen (1987) introduced 12 arm classes (AC) to classify spiral galaxies from the symmetry and continuity of arms. Galaxies which lack bimodal symmetry and have a spiral-like structure composed of small pieces are called flocculent, while those with clear bimodal symmetry and long continuous arms are called grand-design. Since their classification is

5 5 Fig. 1. Arms of logarithmic spiral: lnr tanξ φ. The pitch angle ξ is set to be 10 degree, a typical value of Sb galaxies (Kennicutt 1981). If the rotational direction is same with φ, the arms are leading. Otherwise, trailing.

6 6 in accordance with the orderliness of spiral arms, flocculent galaxies are classified with smaller ACs, while grand-design galaxies are with larger ACs. As they consider galaxies with AC 5-12 as grand-design, about half of spiral galaxies should be grand-design. However, most spirals, which we often call grand-design fall into AC of 9 or 12. In their paper, galaxies with AC of 9 or 12 account for about 30%. Since galaxies with remarkable spiral arms are prevalent as such, we need a mechanism to yield stable spiral structures. Meanwhile, the rotational velocities of most spiral disks are known to be almost constant (e.g., Sofue et al. 1999). Thus, the angular rotation velocity Ω is inversely proportional to radius R. A disk whose Ω is not independent of R, is said to be in differential rotation. In such a condition, any material structures cannot last for a long time. If there is a long bar or stripe in the center of disk, it will be tightly wound in several rotation periods, about a few Gyr. It implies that if once a spiral structure appeared, the differential rotation winds up the arm, and in a short time, it will be too tightly wound to be observable, that is, the disk will just seem to be round and smooth. This difficulty in sustaining the spiral structure is called the Winding Problem. There are a number of possible resolutions for this problem. One of the most well-known and successful theories is the Density Wave Theory proposed by Lin & Shu (1964). This theory explains the spiral structure as a quasi-stationary density wave like a traffic jam. An orbit of materials in the equatorial plane of a galaxy is often described by an epicycle motion, which is like a motion of the Moon viewed from the outside of our solar system. This kind of orbit forms a rosette figure, so that, orbits are not usually closed. However, if viewed from a frame rotating at a certain frequency Ω P, the orbit could be an almost closed ellipse. In the case of the Moon, its orbit is closed as long as we rotate around the Sun at the same frequency with the Earth. If this frame is rigid, i.e. Ω P is independent of R, aligned set of such ellipses can cover a range of radius. As position angles of major axes of ellipses slightly change with radius, orbits are crowded at some part of the disk. In this orbit crowding region, density becomes higher than the remaining disk. This difference in density yields a wave pattern and a spiral arm is formed. Therefore, the spiral pattern is at rest in a frame rotating at Ω P. This frequency, Ω P, is a fundamental parameter in study of galaxies, and is called the Pattern Speed. The shape of this pattern is determined by the underlying gravitational potential of the disk stars, and this potential would be determined by the distribution of old and low-mass stars, since they occupy a large fraction of the total stellar mass. In addition to such a stationary wave, a transitory wave can yield or enhance the spiral structure. A tidal tail caused by a passage of another galaxy is a common structure. A pattern of the central bar is thought to enhance the outer spiral wave pattern. According to the density wave theory, there are two main types of resonances (Binney &

7 7 Tremaine 1987). Around these resonances, physical environments would be greatly different from the rest of the disk. One is called the Corotation Resonance (CR), described as Ω R CR Ω P (2) This means that at the CR, the rotational speeds of materials and the pattern are coincident. The radius R CR, which satisfies this equation is called the Corotation Radius. As the rotation speed of gas in a pattern-rest frame becomes small, a shock wave called the galactic shock would not occur around this resonance. Since galactic-scale star formations would be enhanced by this shock, the star formation is less efficient around the CR. As the relative velocity of materials viewed from the pattern changes its sign at the CR, the direction of streaming motions due to the galactic shock should be different between the inside and outside of the CR. These observable features have been used to locate the CR (See section 1.1.2). Another important resonance is the Lindblad Resonance (LR), which is defined by where m is the number of arms, and κ Ω R LR Ω P 2Ω 1 κ m (3) R dω 2Ω dr is the epicyclic frequency. If the rotation curve is flat, Ω R 1, and then κ 2Ω. If the galaxy rotates rigidly, Ω is constant and κ 2Ω. The signs of plus and minus in equation (3) correspond to the inner and the outer Lindblad resonances (ILR and OLR), respectively. Assuming that the spiral arm is tightly wound, calculation of the dispersion relation indicates that waves in the stellar disk cannot exist across the LRs. Meanwhile, waves in the gaseous disk can pass smoothly through the LRs, since the dispersion relations for the gaseous and stellar disk are different. This calculation states that gases and stars should exhibit different behavior at the LRs. As described above, the pattern speed is one of the most fundamental parameters, which influences the star formation and determines the location and even the existence of resonances. Therefore, its determination is very important to the study of galaxies. However, it cannot be determined directly from observations, since the pattern structure is not a material, but a density wave. (4) Previous Methods Several methods have been proposed to date for determining the pattern speed. The earlier common technique is to locate specific resonances on radii where the property of arms changes.

8 8 Roberts et al. (1975) adopted Ω P for 24 galaxies so that the corotation radius lies nearly coincident with the radial extent of the arms, the easily visible disk, and the distribution of HII regions. Elmegreen et al. (1992) used B-band images to locate 5 resonances for 18 galaxies. Such techniques are, however, subject to the depth of imaging observations. In addition, the choice of tracers should be carefully considered, since stars and gases behave differently at resonances and their distributions are not identical. Cepa & Beckman (1990) derived the star formation efficiency (SFE) in the arm and interarm region for NGC 628 and NGC 3992 from Hα and HI data, and found that the arm-to-interarm ratio of the SFE drops to almost unity at a certain radius. Since star formations at spiral arms are thought to be least efficient around the CR, they concluded that this radius might be the corotation radius. However, it is difficult to derive the SFE values in interarm regions, since there are fewer stars and gases than in arm regions. Canzian (1993) showed that the residual velocity field, obtained by subtracting the axisymmetric component from the observed velocity field, should be different inside and outside the corotation radius. This technique is called the Canzian test. Sempere et al. (1995) applied this 110. Canzian & Allen (1997) applied method to the HI data of NGC 4321, and derived R CR this method to the Hα data of the same galaxy and concluded that the corotation radius should be in the range Since this test needs precise kinematical information over the entire disk, an accurate velocity field with a wide range of radius is required. Tremaine & Weinberg (1984) presented a method, which did not use the morphological location of resonances but use the continuity equation for the surface brightness of galaxies. This method is called the Tremaine-Weinberg (TW) method. Although the TW method has been applied to many galaxies, no observable matters completely obey the continuity equation, since stars are formed from and return to the interstellar gases in a certain period. We should note that a typical value of the SFE is 10 9 yr 1, and that this is not small enough to neglect. The corresponding gas consumption time, SFE 1 of 1 Gyr is just a few rotation periods. Thus, the most successful and reliable applications have been to early type barred galaxies, such as NGC 936 (Merrifield & Kuijken 1995), NGC 4596 (Gerssen et al. 1999), and NGC 7079 (Debattista & Williams 2004), since there are less star formations than in late type galaxies. Gerssen et al. (2003) applied the method to four barred spirals, NGC 271, NGC 1358, ESO , and NGC The resultant value of the latest type galaxy, NGC 3992, is more than twice as large as that of the other galaxies. They thought that this result may be due to the assumption of continuity. In recent studies, HI and CO data have come to be used for gas-dominating, late type galaxies as M81 (Westpfahl 1998), M51 (NGC 5194), M83, and NGC 6946 (Zimmer et al. 2004). Meanwhile, Debattista (2003) showed by the N-body simulation that when this method is applied to a barred galaxy, the result should be sensitive to the uncertainty in adopted position angle (P.A.) of outer disk. Rand & Wallin (2004)

9 9 applied the TW method with varying P.A.s and confirmed that the uncertainty in P.A. made the resultant Ω P with larger error. They also showed that the best-fitted Ω P only with an inner bar was larger than that of the remaining disk, even though a galaxy in interest was not classified as SB galaxy. The results of numerical simulations also show a large dependence of kinematics and spiral structure on Ω P (e.g., Wada et al. 1998). They compared the results of simulations with the observation, and derived the best-fitted value of Ω P. Oey et al. (2003) assumed an evolutional model of HII luminosity function to draw isochrones of massive stars. They fitted these isochrones to the distribution of HII regions, and derived Ω P. These approaches, of course, depend on their modelings, and often have difficulty in estimating the accuracy of the derived values and the effect of other parameters on their results. We list results of the pattern speed determination from previous studies in table 1. Results for our target galaxies, NGC 4254, NGC 4321, and NGC 5194, are excluded from this table, and are listed in table Molecular Clouds and Star Formation Galactic Shock-Wave Theory As we mentioned in section 1.1.1, there is a spiral potential owing to old stars in a spiral disk. Materials such as stars and gases move in this potential at the rotational velocity, and rotate in an almost circular orbit. When gases move into the trough of potential, velocity difference between the potential and gases becomes much larger than the sound speed, so that, a shock wave called the galactic shock occurs. Owing to this shock wave, these gases are accumulated and compressed, and then form dense molecular clouds, which eventually have a core to produce protostars. Thus, star formation is enhanced at spiral arms by the galactic shock. This scenario is described by the Galactic Shock-Wave Theory (Fujimoto 1968; Roberts 1969). Since strong ultraviolet (UV) rays emitted by these massive stars, formed in molecular clouds, dissociate the surrounding molecules into atoms, molecular clouds belonging to a spiral arm are short-lived. Namely, molecular clouds disappear soon after their formation, and their positions appear to be fixed to the spiral pattern, though individual molecular clouds move at the rotational velocity. Massive protostars born at molecular arms also move at the rotational velocity, but need a certain time to be bright, so that, stellar arms should be located at the downstream side of molecular arms. As the lifetimes of massive stars are as short as 10 7 yr, the blue light, which is mainly emitted by massive stars, can delineate arm structures.

10 10 Table 1. Previous results of pattern speed determination Authority Target RC3 Type ΩP RCR Method Data NGC 3031 SA(s)ab kpc morphology Hα NGC 6946 SAB(rs)cd kpc morphology Hα Roberts et al. (1975) Kent (1987) NGC 936 SB(rs) TW Cepa & Beckman (1990) NGC 628 SA(s)c 56 SFE ratio Hα, HI NGC 3992 HI HI kpc simulation K TW CaII SB(rs)bc 32 SFE ratio Hα, HI Elmegreen et al. (1992) NGC 628 SA(s)c 141 morphology B NGC 3031 SA(s)ab 463 morphology B Merrifield & Kuijken (1995) NGC 936 SB(rs) TW Westpfahl (1998) NGC 3031 SA(s)ab TW Bureau et al. (1999) NGC 2915 I TW Gerssen et al. (1999) NGC 4596 SAB(rs)ab TW Kuno et al. (2000) NGC 3504 SAB(rs)ab 41 streaming motion CO Sorai et al. (2000) NGC 253 SAB(s)c morphology CO, HI Weiner et al. (2001) NGC 4123 SB(r)c 20 simulation Hα Gerssen et al. (2003) NGC 271 SB(rs)ab 25 9 TW NGC 3992 SB(rs)bc 73 5 TW Kranz et al. (2003) NGC 3893 SAB(rs)c NGC 5676 SA(rs)bc kpc simulation K Oey et al. (2003) NGC 157 SAB(rs)bc 6 kpc isochrone Hα NGC 6951 SAB(rs)bc 10 kpc isochrone Hα Debattista & Williams (2004) NGC 7079 SB(r)0? Rand & Wallin (2004) NGC 1068 (R)SA(rs)b TW CO NGC 4736 (R)SA(r)ab TW CO Zimmer et al. (2004) NGC 5236 SAB(s)c 45 8 TW CO NGC 6946 SAB(rs)cd 39 8 TW CO 0 0 Since their sample galaxies are more than 2, we just list 2 of them.

11 11 Therefore, the offset between the arms of massive stars and molecular clouds represents difference between the speed of materials and the spiral pattern, as well as the time needed for star formation from molecular clouds (See section 2.1 for the formulation). This time has been estimated to be around 10 7 yr from the calculation of free-fall time, or the Jeans time in molecular clouds Observational Tracers Most molecular gases in galaxies are molecular hydrogen (H 2 ). However, the direct observation of H 2 molecules is very difficult, since an H 2 molecule is symmetric and does not have a dipole moment. Absorption lines in UV or the lowest quadrupole moment emission lines in IR have been observed. Instead of such direct observations of H 2, rotational emission lines of a carbon monoxide (CO), which is the most abundant molecule except for H 2, have been used to derive the amount of molecules as N H 2 X CO I CO K km s 1 cm 2 (5) where N H 2 is the column density of H 2 and I CO is the integrated intensity (!#" T CO dv) of the CO emission. Young & Scoville (1991) calculated the virial masses (M VIR ) of the Galactic molecular clouds from their sizes (a) and velocity widths ( V ) of 12 CO(J=1-0) line as M VIR a V 2$ G. They found that 12 CO(J=1-0) luminosities are proportional to M VIR, and adopted the value of X CO 3 % cm 2 K km s 1 for clouds with M VIR 1 20 % 10 5 M&. This conversion factor is, however, subject to the environments such as the density, temperature, and metallicity. Meanwhile, the amount of star formation has been estimated by various methods. The earlier technique uses the B-band luminosity, since blue lights can be assumed to be dominated by young stars. If we adopt a certain synthesis model, we could calculate the star formation rate (SFR). However, this SFR is prone to systematic errors from an initial mass function (IMF), age, metallicity, and so on. The aforesaid assumption also breaks down in many galaxies. Moreover, the B-band light is very subject to the extinction. Recombination lines such as Hα, Hβ, Pα, Pβ, Brα, and Brγ are the most common tools to derive the SFR. These lines are emitted from HII regions, where interstellar gases surrounding young massive stars are ionized by UV emissions from their central stars. If we assume that all massive stars can be traced by the surrounding ionized gas, we can estimate the SFR from the luminosity of recombination lines. Kennicutt (1983) derived a conversion relation of the SFR and the Hα luminosity as SFR total L Hα 1 12 % erg s 1 M& yr 1 (6)

12 12 assuming the IMF of ψ m m 1' ( m ( 1 M& and ψ m m 2' 5 1 ( m ( 100 M&. Though this formulation has been widely used, the Hα emission is also affected by the extinction. In order to obtain an accurate SFR, we have to consider the extinction due to dust in our Galaxy and in the target galaxy. While the former effect is often small for galaxies with high galactic latitude (Schlegel et al. 1998), the latter is significant but difficult to correct. One way is to use the line ratio of Hα $ Hβ, the Balmer decrement. Since this ratio is theoretically calculated, the amount of the extinction can be estimated from the observed line ratio. However, it is difficult to obtain a map of Hβ, since this line is rather weak and subject to the extinction. Instead of such optical lines, recombination lines with longer wave length, such as Paschen and Brackett lines at NIR, can be used. Though these lines are more resistant to the extinction, they are typically 1-2 orders of magnitude weaker than Hα. Their weakness are the dominant reason why the Hα emission has been used to investigate star formation activities in disk region. The flux of thermal radio continuum can be used to measure the SFR, but the spatial resolution might be not small enough to trace the spiral arm structure. Kennicutt (1998) compiled the ways to calculate the SFR, including some other tracers, such as forbidden lines, and the continuum flux in UV and FIR. Though the extinction is significant to the luminosity of Hα and B, we do not need an accurate luminosity for this work. Since we just pay attention to the position of arms, and the amount of extinction is smaller in nearly face-on galaxies, we did not perform any corrections for extinction. Thus, the offset between molecular and massive stellar arms can be measured by the positions of CO arms and Hα or B-band arms. We will discuss the effect of extinction in section 5.4. A kind of this offset has been found in many spiral galaxies (e.g., Rand & Kulkarni (1990)). We can easily see it in a B-band image as a displacement of an optical arm and a dust lane, since the dust lane appears as a result of extinction due to dense molecular clouds. However, this offset has not yet been studied quantitatively, since the peak position of dust lane is hard to trace and the resolution of CO image was too large to examine the detailed structure of spiral arms. In this paper, we report a new way to determine Ω P and t SF simultaneously, which uses the offset between the arms of massive stars and molecules. In section 2, we introduce a basic concept of the method and some requirements for selection in target galaxies. Properties of sample galaxies and data we used are shown in section 3. In section 4, practical processes for the analysis are described in five steps, and the results and discussions are presented in the following section 5. We give a summary of this work in section 6.

13 13 2. Method 2.1. Basic Idea and Formulation We assume that the pattern of a spiral is rigid, and that the materials rotate in pure circular orbits. The former assumption means that a spiral pattern does exist and its angular speed (Ω P ) is constant. This can be applied to almost all clear or grand-design spiral galaxies, which are thought to sustain quasi-stationary density waves. On the other hand, velocity fields of most galaxies exhibit patterns of a spider diagram, which is a velocity field yielded from pure circular rotations. The streaming motion and the velocity dispersion certainly generate some non-circular motion, but in a spiral disk these are usually about 10 km s 1 (Adler & Westpfahl 1996; Combes & Becquaert 1997). This value is very small compared to the circular rotation velocity of around 200 km s 1 in most disks. These observational results in velocity support the latter assumption. We should note that we do not consider nor include any bar structures in our analysis, since the pattern speed of a bar would be different from that of a spiral (e.g., Wada et al. 1998) and particles within a bar potential move in an elliptical orbit with high eccentricity. Then, we define a timescale, t SF, as an average time for the massive star formation from galactic-shock-compressed molecular clouds. If the physical process of star formation is not extremely changed by the location or environment in the spiral disk, this timescale can be regarded as a constant parameter representing a typical value of the entire disk. Figure 2 illustrates this idea for inside of the CR, where Ω ) Ω P. If we observe a face-on spiral galaxy at t 0 (the left panel), at t t SF the same galaxy will be observed as the right panel and the offset distance between the arm of stars and molecules, d, can be written as d v km s 1 t SF s v P km s 1 t SF s km (7) where v is the velocity of materials, and v P is the velocity of pattern. We adopt this expression, in which d becomes a positive value. This is because most molecules exist in a rather central part of the disk, and thus they are thought to be inside of the CR, while the atomic (HI) gas is distributed in an outer region and often beyond the optical disk. If molecular gas is more extended and outside of the CR should be taken into account, massive stellar arms will be seen on the concave side of molecular arms, and d will be negative. Dividing both sides of equation (7) by radius R [kpc], we obtain θ +* Ω km s 1 kpc 1 Ω P km s 1 kpc 1 -, % t SF s km kpc 1 (8) where Ω! v $ R, Ω P! v P $ R, and θ is the azimuthal offset. This equation shows the relation between

14 14 Fig. 2. Basic idea of our method. If we observe a face-on spiral galaxy at t 0 (the left panel), the same galaxy will be observed as the right panel at t t SF. The thick solid lines are molecular arms at time t of each panel. The thick dashed lines in the right panel (t t SF ) show the position of molecular arms in the left panel (t 0). The offset distance between the massive stars and molecular arm is d, expressed in equation (7).

15 15 two observables, Ω and θ, and we can rewrite it as θ * Ω km s 1 kpc 1 Ω P km s 1 kpc 1 /, % t SF 10 7 yr degree (9) If we assume a constant t SF in a spiral disk, θ becomes a linear function of Ω, since Ω P is assumed to be constant. Therefore, by plotting θ against Ω G and fitting them with a line, both Ω P and t SF can be determined at the same time. In addition, the linearity of this plot can verify the validity of assumptions we made. We discuss this validity in section Requirements for the Application To apply our method to real galaxies, we need to trace both molecular and stellar arms in precise. Nearby galaxies with small inclination will satisfy this requirement. Moreover, we need to obtain the rotational velocity. As the line-of-sight velocities of an almost face-on galaxy do not give velocities parallel to the disk, the inclination angle must not be around zero. Therefore, galaxies with mild inclination are required. The AC is useful to find clear spiral structures. Galaxies with AC of 9 or 12 are preferable. We do not mind whether a galaxy has a bar in its center or not. We just put importance on spiral structures. We selected galaxies by the following criteria. (i)co arms as well as B and Hα arms can be clearly traced. (ii)the AC is 9 or 12, i.e. optically grand-design. (iii)the inclination is larger than 150 and smaller than 500.

16 16 3. Data 3.1. NGC 4254 We observed a SAc galaxy, NGC 4254, in the 12 CO (J 1 0) line using the Nobeyama Millimeter Array (NMA) during a long-term project, Virgo high resolution CO survey (Sofue et al. 2003a). The spatial resolution was % , which corresponds to 230 pc % 180 pc at a distance from the Virgo cluster of 16.1 Mpc (Ferrarese et al. 1996). This resolution is small enough to trace the spiral arms with a typical width of 1 kpc. For more detailed information about the result of observations, see Sofue et al. (2003c). In addition to the interferometric data, single dish data of this galaxy obtained by the Nobeyama 45m telescope was kindly provided by H. Nakanishi, in private communication. This data was used to calculate the rotation curve of the whole molecular disk. Koopmann et al. (2001) presented broadband R and narrowband Hα images of 63 spiral galaxies in the Virgo cluster. The resolution of maps of NGC 4254 is about 1. 2, comparable to that of our CO data. We derived the coordinate of Hα image, using that of Galactic stars seen in the R- band image and a task ccmap of IRAF. The typical uncertainty of this coordinate fitting is about 1, smaller than the resolution of images. We show an overlaid image of CO and Hα in figure 3. The B-band image from the Digitized Sky Survey (DSS) 1 was retrieved from ESO Archive. 2 The overlaid image of CO and B is presented in figure 4. The position angle (P.A.) and inclination (i) are also important parameters of a galaxy. The P.A. is an angle of the major axis of disk from north to east. The i is an angle of how the disk is inclined to the plane normal to the line of sight. A face-on galaxy is expressed by i 00, while an edge-on galaxy is i 900. We determined them by applying a task gal of AIPS to the interferometric CO velocity field. This task cuts annuli out from a velocity field, fits them with a spider diagram, and derives P.A. and i at each radius. We averaged the derived values at 5-3 R 3 16, corresponding to 0.4 kpc 3 R kpc, where the values are almost constant, and then obtained P.A. = and i In table 2, these P.A. and i are listed along with values from other papers. Although only the central region was used in this work, the determined values are not quite different from these given by previous works, in most of which the whole image of the galaxy was used for the determination. Using the obtained P.A. and i, we calculated two rotation curves by means of gal again. One 1 The DSS was produced at the Space Telescope Institute (STScI) and founded by the National Aeronautics and Space Administration. 2

17 17 is derived by the interferometric CO data, so that, this is only for the central region. This rotation curve is drawn by a solid line in figure 5, and is not perfectly coincident with that derived by the iteration method (Takamiya & Sofue 2002; Sofue et al. 2003b), since they used different values of P.A. and i. The other is derived by the single dish CO data to obtain rotational velocities in the whole disk. This rotation curve is drawn by a dashed line in figure NGC 4321 and NGC 5194 The BIMA Survey of Nearby Galaxies (BIMA SONG) is a systematic imaging study in the 12 CO(J=1-0) line for 44 nearby galaxies (Helfer et al. 2003). They used the 10-element Berkeley- Illinois-Maryland Association (BIMA) millimeter interferometer, and performed multifield observations for 33 galaxies, including NGC 4321 and NGC 5194, in order to cover the whole disk Since we adopt D 16 1 Mpc for NGC 4321 and D 9 6 Mpc for NGC 5194, 14 corresponds to 78 pc and 47 pc, respectively. For NGC 4321, they observed 7 fields and the synthesized beam was % 4 2 9, corresponding to 560 pc % 380 pc. For NGC 5194, they observed 26 fields and the synthesized beam was % , corresponding to 220 pc % 190 pc. These images were retrieved from the NASA/IPAC Extragalactic Database (NED). 3 Martin & Kennicutt (2001) presented Hα images of 32 nearby galaxies, including our sample galaxies. As they kindly gave us their Hα and R-band data, we could derive the coordinate of these images in the same way as NGC We retrieved B-band images of the DSS from ESO Archive. The overlaid images of CO and Hα or B are presented in figure 3 or 4, respectively. Sofue et al. (1999) presented high-resolution rotation curves of 50 spiral galaxies, including the Milky Way. We used these rotation curves (figure 5) and parameters they adopted except for Table 2. Position angle and inclination of NGC Authority P.A. [6 ] i [6 ] 7 7 Schweizer (1976) Iye et al. (1982) Phookun et al. (1993) This work (2005)

18 18 the distance of NGC We list parameters of these three galaxies in table 3.

19 Table 3. Sample Galaxies 59 ] ] FWHM [ ] D [Mpc] AC pitch angle [ ] i [ Name RA (J2000) Dec (J2000) RC3 Type PA [ 34 9$ 1$ NGC 4254 (M 99) SA(s)c NGC 4321 (M 100) SAB(s)bc 146 # 27 # NGC 5194 (M 51) SA(s)bc pec 22 # 20 # 9.6 # Sofue et al. (2003c) Ferrarese et al. (1996) García Gómez & Athanassoula (1993) Positions from NED # Sofue et al. (1999) $ Helfer et al. (2003)

20 20 Fig. 3. Projected images of sample galaxies. The horizontal and the vertical axes are in Right Ascension (J2000) and in Declination (J2000), respectively. CO contours are superposed on Hα images and crosses indicate the center position of each galaxy.

21 21 Fig. 4. Same as figure 3, but CO contours on B-band images.

22 22 Fig. 5. Rotation curves of sample galaxies. The solid and dashed lines of NGC 4254 are calculated by a task gal of AIPS with the CO velocity fields of interferometric (NMA) and single dish (45m) data, respectively. Rotation curves of NGC 4321 and NGC 5194 are provided by Sofue et al. (1999).

23 23 4. Process of the Application 4.1. Phase Diagram To deproject the image of galaxies into a face-on view, we first rotated a projected image by -P.A., which made a major axis parallel to the columns. Then, we stretched the rotated image along rows (i.e., a minor axis) by 1 $ cosi. These procedures provided a face-on view of galaxies, whose major and minor axis were parallel to the columns and rows. We transformed this deprojected image into polar-coordinate to make a phase diagram (figure 6-11). A kind of this diagram was first presented by Iye et al. (1982) as the logarithmic polar coordinates. In our phase diagram, abscissa is azimuthal angle and ordinate is radius in a common logarithms. The azimuth is set to be zero at minor axis, and to increase with the each galaxy s rotation, assuming trailing arms. This azimuth corresponds to φ in figure 1, so that, equation (1) should be rewritten as log 10 R tanξ log 10 e φ (10) Thus, in the phase diagrams, logarithmic spiral arms can be readily recognized by straight lines with a gradient of tanξ log 10 e. We could find such lines and estimated pitch angles ξ for each galaxy Azimuthal Intensity Profile We divided the phase diagram into strips of 200 pc (NGC 4254 and NGC 4321) or 100 pc (NGC 5194) radial width. This width is roughly consistent to the resolution of CO data for each galaxy. We then averaged the intensity with respect to the radius at each azimuth in each strip, and plotted the averaged intensity against azimuth. We defined the offset angle, θ, as an azimuthal angular separation of intensity peaks between the arm of stars and molecules. Peak positions of Hα and B are well consistent in NGC 4254 and NGC 5194, while in NGC 4321, peaks of B arms are slightly shifted to the downstream side from those of Hα arms. Thus, we could derive some offset values from CO and B arms, though there were no offsets between CO and Hα arms. We concluded that the offsets between CO and B arms were less reliable from the effect of extinction, and that offset values should be derived from CO and Hα arms (See section 5 for this discussion). Hence, we did not apply further analyses for NGC Since CO and Hα arms consist of small and clumpy components, we could not find corresponding peaks or offsets at some radii. In addition, we only used the region where the spiral structure is clear and the rotation curve does not abruptly change with radius. In figure 6-11, this

24 24 region is indicated by two solid horizontal lines. The peaks of CO and Hα arms are marked by filled circles and boxes, respectively. Their labels correspond to those in figure θ Ω Plot In figure 12, we plotted the derived offsets θ against angular rotation velocities of gas (Ω), which were calculated from rotation velocities (figure 5) divided by radius R. As the rotation curves are almost flat, points near the top right corner come from data with small R. We confirmed the linear relation between θ and Ω form this plot. This linearity can justify the assumptions we made Errors The largest factor for error in θ is the resolution of CO data, which is about 3 for NMA and about 6 for BIMA SONG. This error for a galaxy with the distance of D Mpc and with the resolution of δ arcsec, should be written as θ D Mpc δ arcsec R kpc 1 deg (11) In addition, uncertainty of the coordinate fitting of Hα images, and the fact that the determination of offset angles is somewhat subjective would be small but possible sources of error in θ. We, however, neglected these errors for simplicity of calculations, so that, the error in θ could be larger than the value obtained from equation (11) by a factor of 2, at most Fitting In figure 12, we added errorbars calculated from equation (11) to the θ Ω plot. We fitted a line to each plot by means of the χ-square fitting method. The fitted line is drawn by a red line, and the gradient and horizontal-axis-intercept of this line correspond to the resultant value of t SF and Ω P, respectively.

25 25 Fig. 6. Phase diagram of NGC 4254: CO contours on Hα image. Thick dashed lines indicate rough positions of two marked spiral arms. The region between the two horizontal lines (1 kpc 3 R kpc) indicates where we used in the analysis. Filled circles and boxes are the peaks of CO arm and Hα arm, respectively, which are defined by the analysis described in section 4.2. Labels are to identify these peaks and offsets presented in figure 12.

26 26 Fig. 7. Same as figure 6, but CO contours on B-band image.

27 27 Fig. 8. Phase diagram of NGC 4321: CO contours on Hα image. Thick dashed lines indicate rough positions of spiral arms connected with a central bar. An azimuthal gap between the bar and spirals can be seen at 3 kpc 3 R 3 4 kpc.

28 28 Fig. 9. Same as figure 8, but CO contours on B-band image.

29 29 Fig. 10. Phase diagram of NGC 5194: CO contours on Hα image. Thick dashed lines indicate rough positions of spiral arms. The two solid horizontal lines show where we used in the analysis (2.2 kpc 3 R kpc). The dotted-dashed line at R 1 3 kpc shows the lowest limit of where we could find offsets, which are less reliable. Filled circles and boxes are the peaks of CO arm and Hα arm, respectively, which are defined by the analysis described in section 4.2. Some of them cannot be seen, since they are overlapped. Offsets with label of small letter were not included in the analysis.

30 30 Fig. 11. Same as figure 10, but CO contours on B-band image.

31 31 Fig. 12. Plot of offset angles from CO and Hα data against angular rotation velocities. Labels are to identify where these offsets came from. As the rotation curves are almost flat, points near the top right corner came from data with small R. The errorbars were calculated from equation (11) and the red line is the best-fitted line obtained by the χ-square method. The horizontal-axisintercept and gradient of this line correspond to Ω P and t SF, respectively. For NGC 5194, the fitted line with all offsets, including less reliable offsets with small letter labels, is drawn by the orange dashed line.

32 32 5. Results 5.1. NGC 4254 In the phase diagram (figure 6, 7), we can recognize two clear spirals and a structure like a part of a ring at R 2 kpc. The pitch angles of two arms are slightly different as 250 and 280, in rough consistency with the value of García Gómez & Athanassoula (1993). We used the data of 1 kpc 3 R kpc, indicated by two horizontal lines in the phase diagrams. We derived ten offset values, indicated by labels A-J in figure 6, at every 200 pc in radius, and adopted δ 3 from the CO resolution. The typical value of error in offset is about 9 degree. We found that though B arms are broader than Hα arms, the peak positions of Hα and B are well consistent to each other. This might confirm that the extinction is not significant enough to shift the apparent position of optical arms in this galaxy. The breadth of B arms suggests to us that the B-band luminosity is contaminated by older stars to some extent, and that results from CO and B offsets could be less reliable. Therefore, we used offsets between the arms of CO and Hα, and obtained t SF Myr and Ω P km s 1 kpc 1. We calculated Ω κ and Ω κ $ m to locate resonances (figure 13). From the Ω curve calculated by the single dish data (the black dashed line of figure 13), the corotation radius should be R CR 6 2 kpc or 79. This radius corresponds to where the south arm seems to change its pitch angle (figure 14). Meanwhile, the HI disk of this galaxy by Cayatte et al. (1990) shows a sharp cutoff in the south, which indicates that the outer structure might be influenced by the intracluster medium of the Virgo cluster. Thus, whether this break has been yielded by the internal kinematics of this galaxy, or the external environments of the cluster, is not certain. Locating the Lindblad resonances is rather difficult, since this galaxy has three arms in outer disk, while the third arm is not clear in the central region we used. We calculated frequencies Ω κ $ m with m 1 2 and 3. Taking m 2 mode as a representative in inner region, the Ω κ $ 2 curve form the interferometric data and derived Ω P locate the oilr on about 1 kpc. Though the curve is not smooth enough, this radius coincides where the spiral arm structures can be traced clearly. The OLR is estimated to be around or farther than 9 kpc from the calculation of Ω κ $ m with m 1 2 and 3. Since κ becomes smaller in outer region, the Ω κ $ m curve does not vary greatly with different m. We list previously derived Ω P and/or R CR for this galaxy in table 4. Our result is well consistent to Kranz et al. (2001).

33 33 Fig. 13. Rotation curves and frequencies for NGC The solid and dashed lines are calculated from the CO velocity fields of interferometric (NMA) and single dish (45m) data, respectively. Table 4. Previous derived pattern speeds of NGC 4254 Authority D [Mpc] P.A. [6 ] i [6 ] Ω P R CR Method Data Elmegreen et al. (1992) 87: : morphology B Kranz et al. (2001) : : simulation K, Hα This work (2005) ; 17 < 8 79: : offset CO, Hα

34 34 Fig. 14. The corotation of R CR 79 is superposed on the Hα image of NGC The horizontal bar on the upper side corresponds to 5kpc.

35 NGC 4321 As seen in the overlaid images (figure 3, 4, 8, and 9), CO arms are much weaker than the nuclei and shorter than optical arms. The phase diagrams show that both molecular and stellar arms in outer disk (R ) 4 kpc) can be expressed by a logarithmic function with a pitch angle of 210. However, the central bar exits only within 3 kpc. This inconsistency in the end position of arms and bar suggests that there are different patterns in the center and outer disk. The weakness of molecular arms could be understood by the existence of the central strong bar, which is thought to enhance a large-scale mass inflow. Though we could find some offsets from CO and B-band data, there seemed to be few offsets in CO and Hα data. We think that offsets between CO and B might come from the extinction. If the intrinsic peaks of molecular arms and massive stellar arms are coincident, the amount of extinction becomes larger and the B-band luminosities are suppressed only at where molecular arms exist. In such a case, a pseudo offset will be apparent. Our result of no offsets in CO and Hα arms indicates the following possibilities; (i) offsets are just small compared to the resolution of pc, (ii) the outer arms of this galaxy are material arms, or (iii) orbits cannot be assumed as circles. We list the previous determinations of pattern speed in table 5. The corotation radius of about kpc, corresponding Ω P 27 km s 1 kpc 1 is supported by Elmegreen et al. (1992), Sempere et al. (1995), Canzian & Allen (1997), and Rand & Wallin (2004). Substituting this Ω P and t SF 4 Myr in equation (9), offsets around 70 should be seen at radius of 4 kpc. This value corresponds to 490 pc, which is slightly larger than the resolution. Thus, if the corotation is located around 5 kpc, offsets are too small to observe. In addition, our result of no apparent offsets might be explained if t SF is as small as 0 5 Myr. However, this is less possible condition, since we obtained t SF 4 Myr for the remaining two galaxies. The existence of material arms might be also rejected by the shape of arms, which are well expressed by a logarithms. As seen in the velocity field by Helfer et al. (2003), non-circular motion becomes larger in arm positions. The effect of such motions on offset values is our future subject NGC 5194 In the phase diagrams (figure 10 and 11), we can recognize two clear logarithmic spirals with a single pitch angle of 200. This value is larger than that of 120 by García Gómez & Athanassoula (1993). The inner ends of Hα arms are not so strong as CO arms. From the projected images (figure 3, 4) as well as the phase diagrams, we found that offsets between molecules and stars could be clearly seen, and the peak positions of arms were consistent in Hα and B. This indicates

36 36 that extinction in this galaxy does not affect the apparent positions of optical arms. We averaged intensities at every 100 pc in radius, and adopted δ 6 from the CO resolution, giving the typical error in θ of 5 degree. We derived t SF Myr and Ω P km s 1 kpc 1, with 22 offset values between the arms of CO and Hα. We used data from 2.2 kpc 3 R kpc, where both of CO and Hα arms were clearly traced. This region is indicated by two solid horizontal lines in the phase diagram. Offsets used in this analysis are labeled by large letters from A to V. If we include data from 1.3 kpc 3 R kpc, where the positions of Hα arms 3 were not clear, the resultant values will be t SF Myr and Ω P 15 2 km s 1 kpc 1. The additional 6 offsets are labeled by small letters from a to f. In figure 15, we present the results of calculation of Ω κ and Ω κ $ m with m 2 for NGC 4321 and NGC From the derived Ω and Ω P, the corotation radius should be around 10 kpc or This is over the kink of arms and near the companion galaxy, NGC 5195 (figure 16). From the calculation of κ, the oilr should be over 4kpc. This is not consistent to that we can see spiral arms clearly at R ) 2 kpc. If the non-circular motion is large enough to make offsets smaller, the derived value of Ω P is the lower limit. Though larger Ω P can resolve the inconsistency, we need further discussions. In outer (R ) 8 kpc) region, the calculation of κ yields inaccurate result (κ immediately fell into zero at some radii), probably because the rotational velocity decreases with radius. This peculiar rotation curve might be due to perturbation from the companion galaxy. We list previous results of pattern speed determination in table 6. Many works suggested R CR 120= 160, or 5.6 kpc to 7.5 kpc. This is near the position of where both arms change their pitch angle. However, Kuno et al. (1995) and Oey et al. (2003) derived the corotation radius at near the companion. Our result falls between these two values. Table 5. Previous derived pattern speeds of NGC 4321 Authority D [Mpc] P.A. [6 ] i [6 ] Ω P R CR Method Data Elmegreen et al. (1992) 118: : morphology B Sempere et al. (1995) : : Canzian, simulation HI Canzian & Allen (1997) : : Canzian Hα Wada et al. (1998) (bar) simulation CO Oey et al. (2003) : : isochrone Hα Rand & Wallin (2004) ; 4 < 5 TW CO

37 37 Fig. 15. Rotation curves and frequencies for NGC 4321 and NGC 5194.

38 38 Fig. 16. The corotation of R CR 210 is superposed on the Hα image of NGC The horizontal bar on the upper side corresponds to 5kpc.

39 Discussion We determined t SF 4 Myr for NGC 4254 and NGC This may indicate that the dominant process of star formation from molecular clouds is the gravitational collapse, since the Jeans time in molecular clouds has been calculated as 10 Myr. As this parameter might be depend on the environments such as metallicity, our result suggests that this dependency is originally small, or that these two galaxies are similar in the physical conditions. Including less reliable offsets in NGC 5194 gave slightly different values for both t SF and Ω P. This implies that we need to modify the way of deriving offsets to obtain more accurate results. In our method, we assumed a pure circular rotation and a constant timescale needed for star formation from molecular clouds, and did not make any corrections for the extinction in optical data. As we mentioned in section 2.1, the deviation from the former assumption is small in spiral disks. However, we need to consider the effect of non-circular motion on offset values, since this is one of possible causes for no offsets in NGC If we know how the non-circular motion affects the offsets, our method will be more sophisticated and reliable. The validity of the latter assumption, a constant t SF, can be checked by the linearity of θ Ω plot (figure 12). Since this plot is well fitted with a line, dependence of t SF on the environments should be small and included in the error of derived values. The extinction in optical light is surely significant to the luminosity, and this might lead us to overestimate the offset values. The amount of extinction becomes larger as HII regions or stars become closer to molecular arms, so that, the apparent peak of stellar arms could shift to the downstream side from where it really is. To estimate the amount of this effect, we compared peak positions of Hα and B arms. The B-band light suffers from much more extinction than Hα. Thus, the difference in peak positions of Hα and B can be interpreted as the effect of extinction. In NGC 4321, B-band arms were slightly shifted to the downstream side of Hα arms, whose positions are almost the same with CO arms. This led us to conclude that the extinction in this galaxy might cause the apparent positions of B arms to be shifted. In NGC 4254 and NGC 5194, the peak positions of Hα and B arms were coincident, so that, we could conclude that the extinction in these galaxies hardly change the positions of optical arms, and neglect the effect of extinction on the offset values. In addition, we discuss about the effects of error in the adopted parameters on resultant values. Since the rotation curve is almost flat, the angular rotational velocity, Ω is inversely proportional to the distance D. Thus the derived t SF is proportional to D, but the derived Ω P is independent. The error in offset angle is biggest around the major axis and smallest around the minor axis, since we extended images by 1 $ cosi along the minor axis. The largest error is about 20% for NGC 4254, which is the most inclined galaxy in our sample, from the uncertainty in i of 50.